## A justification principle about logical implication

[Update 11/22/09]

I note that Ben actually talked about this principle in a post on his blog, “if it’s reasonable to believe a bunch of premises, it’s also reasonable to (on the basis of the logical connection) believe the conclusions that can be validly inferred from those premises”,

[/update]

I have recently been discussing Gettier’s famous counter-examples to the JTB theory of knowledge. In his original paper Gettier argued that there are some cases where all the necessary and sufficient conditions of knowledge according to JTB theory are met, but the person in question fails to know. In the thread user ACB asked that:

If (1) the man who will get the job is Jones, and
(2) Jones has ten coins in his pocket,
then
(3) the man who will get the job has ten coins in his pocket.

But does it logically follow that if Smith is justified in believing (1) and (2), then he is justified in believing (3)? [followed by a proposed counter-example]

I and another person thought that it did follow. In other words we subscribed to the following principle about justification:

For all persons, for all propositions, P, and for all propositions, Q, that a person is epistemically justified in believing that P, and that P logically implies Q logically implies that that person is epistemically justified in believing that Q.
(∀x)(∀P)(∀Q)(Jx(P)∧P⇒Q))⇒Jx(Q)

The above case seems to me to be a true instantiation of the justification principle. ACB disagreed with the principle and proposed a counter-example with the alphabet which did not convince me. He then tried another counter-example that involved some mathematical propositions. That proposed counter-example did not convince me either, but it did make me think of an example that did convince me. Here’s my counter-example:

1. 1+1=2

2. 456·789=359784

Both of these propositions are true, they are even necessarily true. According to the definition of logical implication they imply each other (and themselves), since any necessarily true proposition imply any (other) necessarily true proposition.1

Now suppose that a child is learning elemental math. Say that she has not even learned multiplication yet, however she has learned that 1+1=2 is true and she knows this. That implies that she is epistemically justified in her belief that 1+1=2. But it clear to me that she is not epistemically justified in believing that 456·789=359784. This is a counter-example to the justification principle and the principle is therefore false.

It seems to me that one could perhaps save the justification principle with some relevance logic understanding of “logical implication”. However I shall not pursue that here.

### Notes

1The definition of “logical implication” is: a proposition logically implies another proposition iff in all possible worlds where the first proposition is true, so is the second.